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Theorem rankunb 8751
Description: The rank of the union of two sets. Theorem 15.17(iii) of [Monk1] p. 112. (Contributed by Mario Carneiro, 10-Jun-2013.) (Revised by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
rankunb ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))

Proof of Theorem rankunb
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unwf 8711 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) ↔ (𝐴𝐵) ∈ (𝑅1 “ On))
2 rankval3b 8727 . . . . . . 7 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
31, 2sylbi 207 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦})
43eleq2d 2716 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ 𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦}))
5 vex 3234 . . . . . 6 𝑥 ∈ V
65elintrab 4520 . . . . 5 (𝑥 {𝑦 ∈ On ∣ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦} ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦))
74, 6syl6bb 276 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) ↔ ∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦)))
8 elun 3786 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
9 rankelb 8725 . . . . . . . . 9 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ (rank‘𝐴)))
10 elun1 3813 . . . . . . . . 9 ((rank‘𝑥) ∈ (rank‘𝐴) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
119, 10syl6 35 . . . . . . . 8 (𝐴 (𝑅1 “ On) → (𝑥𝐴 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
12 rankelb 8725 . . . . . . . . 9 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ (rank‘𝐵)))
13 elun2 3814 . . . . . . . . 9 ((rank‘𝑥) ∈ (rank‘𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
1412, 13syl6 35 . . . . . . . 8 (𝐵 (𝑅1 “ On) → (𝑥𝐵 → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1511, 14jaao 530 . . . . . . 7 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((𝑥𝐴𝑥𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
168, 15syl5bi 232 . . . . . 6 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (𝐴𝐵) → (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
1716ralrimiv 2994 . . . . 5 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))
18 rankon 8696 . . . . . . 7 (rank‘𝐴) ∈ On
19 rankon 8696 . . . . . . 7 (rank‘𝐵) ∈ On
2018, 19onun2i 5881 . . . . . 6 ((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On
21 eleq2 2719 . . . . . . . . 9 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((rank‘𝑥) ∈ 𝑦 ↔ (rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2221ralbidv 3015 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
23 eleq2 2719 . . . . . . . 8 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → (𝑥𝑦𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2422, 23imbi12d 333 . . . . . . 7 (𝑦 = ((rank‘𝐴) ∪ (rank‘𝐵)) → ((∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) ↔ (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
2524rspcv 3336 . . . . . 6 (((rank‘𝐴) ∪ (rank‘𝐵)) ∈ On → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵)))))
2620, 25ax-mp 5 . . . . 5 (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) → (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ ((rank‘𝐴) ∪ (rank‘𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2717, 26syl5com 31 . . . 4 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (∀𝑦 ∈ On (∀𝑥 ∈ (𝐴𝐵)(rank‘𝑥) ∈ 𝑦𝑥𝑦) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
287, 27sylbid 230 . . 3 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (𝑥 ∈ (rank‘(𝐴𝐵)) → 𝑥 ∈ ((rank‘𝐴) ∪ (rank‘𝐵))))
2928ssrdv 3642 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) ⊆ ((rank‘𝐴) ∪ (rank‘𝐵)))
30 ssun1 3809 . . . . 5 𝐴 ⊆ (𝐴𝐵)
31 rankssb 8749 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐴 ⊆ (𝐴𝐵) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵))))
3230, 31mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐴) ⊆ (rank‘(𝐴𝐵)))
33 ssun2 3810 . . . . 5 𝐵 ⊆ (𝐴𝐵)
34 rankssb 8749 . . . . 5 ((𝐴𝐵) ∈ (𝑅1 “ On) → (𝐵 ⊆ (𝐴𝐵) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵))))
3533, 34mpi 20 . . . 4 ((𝐴𝐵) ∈ (𝑅1 “ On) → (rank‘𝐵) ⊆ (rank‘(𝐴𝐵)))
3632, 35unssd 3822 . . 3 ((𝐴𝐵) ∈ (𝑅1 “ On) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
371, 36sylbi 207 . 2 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → ((rank‘𝐴) ∪ (rank‘𝐵)) ⊆ (rank‘(𝐴𝐵)))
3829, 37eqssd 3653 1 ((𝐴 (𝑅1 “ On) ∧ 𝐵 (𝑅1 “ On)) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 382  wa 383   = wceq 1523  wcel 2030  wral 2941  {crab 2945  cun 3605  wss 3607   cuni 4468   cint 4507  cima 5146  Oncon0 5761  cfv 5926  𝑅1cr1 8663  rankcrnk 8664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-r1 8665  df-rank 8666
This theorem is referenced by:  rankprb  8752  rankopb  8753  rankun  8757  rankaltopb  32211
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